A significant consideration which must be faced by financial institutions (and individual investors) is the potential risk of future losses which is inherent in a given financial position, such as a portfolio. There are various ways for measuring potential future risk which are used under different circumstances. One commonly accepted measure of risk is the value at risk (“VAR”) of a particular financial portfolio. The VAR of a portfolio indicates the portfolio's market risk at a given percentile. In other words, the VAR is the greatest possible loss that the institution may expect in the portfolio in question with a certain given degree of probability during a certain future period of time. For example, a VAR equal to the loss at the 99th percentile of confidence level indicates that there is only a 1% chance that the loss will be greater than the VAR during the time frame of interest.
Generally, financial institutions maintain a certain percentage of the VAR in reserve as a contingency to cover possible losses in the portfolio in a predetermined upcoming time period. It is important that the VAR estimate be accurate. If an estimate of the VAR is too low, there is a possibility that insufficient funds will be available to cover losses in a worst-case scenario. Overestimating the VAR is also undesirable because funds set aside to cover the VAR are not available for other uses.
To determine the VAR for a portfolio, one or more models which incorporate various risk factors are used to simulate the price of each instrument in the portfolio a large number of times using an appropriate model. The model characterizes the price of the instrument on the basis of one or more risk factors, which can be broadly considered to be a market factor which is derived from tradable instruments and which can be used to predict or simulate the changes in price of a given instrument. The risk factors used in a given model are dependent on the type of financial instrument at issue and the complexity of the model. Typical risk factors include implied volatilities, prices of underlying stocks, discount rates, loan rates, and foreign exchange rates. Simulation involves varying the value of the risk factors in a model and then using the model to calculate instrument prices in accordance with the selected risk factor values. The resulting price distributions are aggregated to produce a value distribution for the portfolio. The VAR for the portfolio is determined by analyzing this distribution.
There are two alternative simulation techniques which are conventionally used during risk analysis, such as VAR calculations: parametric simulation and historical simulation.
In a parametric simulation, the change in value of a given price for a security is simulated by changing the value of the risk factors in the model from their initial values according to a stochastic or random function. A well known model used in option pricing is the Black-Scholes model which models the change in a stock price S over a time interval t as a function of σ √{square root over (Δ/ε)}, where σ is a risk factor indicating the volatility of the price, and ε is a random component. Parametric simulation has the advantage of being very flexible. For example, the values of the parameters which define the model can be adjusted as required to make the model risk neutral. In addition, when the starting values of the model parameters cannot be determined or implied from actual data, default parameters can be used until reliable historical or market data is available.
A serious drawback to this technique, however, is that the noise components ε used to vary the risk factor values are generally assumed to have a normal distribution. In reality, low probability events occur with more frequency than in a normal distribution. As a result, so-called “fat-tails” of the probability curve must be explicitly defined in the model and used to alter the normal distribution of ε.
Another problem with parametric models is that the model must expressly model cross-correlations between various risk factors. Typically, a variance-covariance matrix is used to preserve a predetermined correlation between the various risk factors during a simulation. An underlying assumption to this technique is that the correlations between various factors are constant across the range of input parameters. However, the correlations can vary depending on the circumstances. Detecting these variations and compensating for them through the use of multiple variance-covariance matrices is difficult and can greatly complicate the modeling process. In addition, the computational cost of determining the cross-correlations grows quadratically with the number of risk factors. It is not unusual for large derivative portfolios to depend on 1000 or more risk factors and determining the cross-correlations for the risk factors quickly becomes unmanageable, particularly when the simulation process must be run daily.
An alternative to parametric simulation is historical simulation. In a historical simulation, a historical record of data is analyzed to determine the actual risk factor values. To simulate price evolution, risk factor values are selected at random from the historical set and applied to the model to determine the next price in the simulation. This approach is extremely simple. Because historical data is used as a direct source for the risk factor values, the methodology does not require calculation of model parameters, such as correlations and volatilities. Moreover, the fat-tail event distribution and stochastic correlations between various factors is automatically reproduced. However, this method is limited because the statistical distribution of values is restricted to the specific historical sequence which occurred. In addition, historical data may be missing or non-existent, particularly for newly developed risk factors, and the historical simulation is generally not risk neutral.
Accordingly, there is a need for an improved technique for adjusting the value of risk factors during simulation of a financial instrument, e.g., for use in risk analysis.